37. If a be taken twice, three times, &c., the results are represented in algebra by 2a, 3a, 4a, &c. The sum of any two of this series may be expressed in a shorter form than by writing the sign + between them; for though we do not know what number a stands for, we know that, be it what it may, 2a + 2a = 4a, 3a + 2a = 5a, 4a + 9a = 13a; and generally, if a taken m times be added to a taken n times, the result is a taken m + n times, or
ma + na = (m + n)a.
38. The use of the brackets must here be noticed. They mean, that the expression contained inside them must be used exactly as a single letter would be used in the same place. Thus, pa signifies that a is taken p times, and (m + n)a, that a is taken m + n times. It is, therefore, a different thing from m + na, which means that a, after being taken n times, is added to m. Thus (3 + 4) × 2 is 7 × 2 or 14; while 3 + 4 × 2 is 3 + 8, or 11.
39. When one number is taken away from another, the number which is left is called the difference or remainder. The process of finding the difference is called subtraction. The number which is to be taken away must be of course the lesser of the two.
40. The process of subtraction depends upon these two principles.
I. The difference of two numbers is not altered by adding a number to the first, if you add the same number to the second; or by subtracting a number from the first, if you subtract the same number from the second. Conceive two baskets with pebbles in them, in the first of which are 100 pebbles more than in the second. If I put 50 more pebbles into each of them, there are still only 100 more in the first than in the second, and the same if I take 50 from each. Therefore, in finding the difference of two numbers, if it should be convenient, I may add any number I please to both of them, because, though I alter the numbers themselves by so doing, I do not alter their difference.
- II. Since 6 exceeds 4 by 2,
- and 3 exceeds 2 by 1,
- and 12 exceeds 5 by 7,
6, 3, and 12 together, or 21, exceed 4, 2, and 5 together, or 11, by 2, 1, and 7 together, or 10: the same thing may be said of any other numbers.
41. If a, b, and c be three numbers, of which a is greater than b (40), I. leads to the following,
(a + c) - (b + c) = a - b.